A Markov Chain on Permutations which Projects to the PASEP

Abstract

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of N sites. It is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. Additionally, particles may enter from the left with probability α and exit to the right with probability β. It has been observed that the (unique) stationary distribution of the PASEP has remarkable connections to combinatorics–see for example the papers of Derrida et al [7, 8], Brak and Essam [1], Brak et al [2], Duchi and Schaeffer [10], Corteel [4], and Shapiro and Zeilberger [15]. Most recently we proved [6] that in fact the (normalized) probability of being in a particular state of the PASEP can be viewed as a certain weight-generating function for permutation tableaux of a fixed shape. (This result implies the previous combinatorial results.) However, our proof relied on the matrix ansatz of Derrida et al [8], and hence did not give an intuitive explanation of why one should expect the steady state distribution of the PASEP to involve such nice combinatorics. In this paper we define a Markov chain–which we call the PT chain–on the set of permutation tableaux which projects to the PASEP, in a sense which we shall make precise. This gives a new proof of the main result of [6] which bypasses the matrix ansatz altogether. Furthermore, via the bijection of [16], the PT chain can also be viewed as a Markov chain on the symmetric group. Another nice feature of the PT chain is that it possesses a certain symmetry which extends the particle-hole symmetry of the PASEP.